%0 Journal Article
%T On Exact Sequence of Semimodules over Semirings
%A Jayprakash Ninu Chaudhari
%A Dipak Ravindra Bonde
%J ISRN Algebra
%D 2013
%R 10.1155/2013/156485
%X We introduce the notion of exact sequence of semimodules over semirings using maximal homomorphisms and generalize some results of module theory to semimodules over semirings. Indeed, we prove, ˇ°If is a split exact sequence of -semimodules and maximal -semimodule homomorphisms with is a strong subsemimodule of , then ˇ±. Also, some results of commutative diagram of -semimodules and maximal -homomorphisms having exact rows are obtained. 1. Introduction The concept of semiring was introduced by Vandiver [1] in 1934. A semiring is a nonempty set together with two associative binary operations, addition and multiplication, which is called a semiring if (i) addition is a commutative operation, (ii) there exists such that , for each , and (iii) multiplication distributes over addition both from left and right. Denote the sets of all nonnegative and residue classes of integers modulo , respectively, by and . The set is a semiring under usual addition and multiplication of nonnegative integers, but it is not a ring. Concepts of commutative semiring, semiring with identity , can be defined on the similar lines as in rings. All semirings in this paper are assumed to be commutative with identity . Generalizing the notion of exact sequence of modules over rings, Abuhlail [2], Al-Thani [3], and Bhambri and Dubey [4] introduced different notions of exact sequence of semimodules over semirings. In this paper, we introduce the notion of exact sequence of semimodules over semirings using maximal homomorphisms, and hence, some results of commutative diagram of -semimodules and maximal -homomorphisms having exact rows are obtained. The following definitions and results will be used to prove our results. A left -semimodule is a commutative monoid with additive identity for which we have a function , defined by and called scalar multiplication, which satisfies the following conditions for all elements and of and all elements and of : (1) ; (2) ; (3) ; (4) ; (5) . Throughout this paper, by an -semimodule we mean a left semimodule over a semiring . Every semiring is a -semimodule (see [5, page 151]). An element of a monoid is called idempotent if . If is an idempotent commutative monoid, then is a -semimodule with scalar multiplication defined by if and if for all and (see [5, page 151]). A nonempty subset of an -semimodule is called a subsemimodule of if is closed under addition and closed under scalar multiplication. A subsemimodule of an -semimodule is called (1) a subtractive subsemimodule (= -subsemimodule) if , , , then ; (2) a strong subsemimodule if for any there
%U http://www.hindawi.com/journals/isrn.algebra/2013/156485/